A permutomino of size n is a polyomino determined by
particular pairs (π1, π2)
of permutations of size
n, such that
π1(i) ≠ π2(i)
for 1 ≤ i ≤ n. Here we determine the
combinatorial properties and,
in particular,
the characterization for the pairs of permutations defining convex
permutominoes.

Using such a characterization, these permutations can be uniquely
represented in terms of the so-called square permutations,
introduced by Mansour and Severini. We provide a closed
formula for the number of these permutations with size n.